Chapter 12, Problem 74A

To determine

Minimum amount of ice needed to cool.

To identify: Whether it is better to cool the tea to room temperature before adding sugar or not.

Answer to Problem 74A

  $m=1.1\text{kg}$

Yes, it is to cool the tea to room temperature before adding sugar.

Explanation of Solution

Given:

Volume of tea, $V=1\text{L}$

Initial temperature, $T_1=0°\text{C}$

Final temperature, $T_2=90°\text{C}$

Formula Used:

Specific heat formula,

  $Q=mC\Delta T$

Where, $Q$ is heat energy, $m$ is mass, $C$ is specific heat, and $\Delta T$ is change in temperature.

Heat fusion formula,

  $Q=m{H}_f$

Where, $Q$ is heat energy, $m$ is mass and $H_f$ is heat of fusion.

Calculation:

The specific heat of water is $C=4180\text{J/kg}\cdot \text{K}$ and

Heat of fusion of ice is $H_f=3.34\times {10}^5\text{J/kg}\cdot \text{K}$

The density of water is around $1{\text{kg/m}}^3$ so, in 1 L mass is 1 Kg.

Now,

The heat that must be lost by the tea is,

  $\begin{array}{l}Q=1\times 4180\times \left(90°-0°\right)\\ Q=3.76\times {10}^5\text{J}\end{array}$

This heat is absorbed by ice as it melts. On putting the values in heat fusion equation,

  $\begin{array}{c}Q=m{H}_f\\ 3.76\times {10}^5=m\times 3.34\times {10}^5\\ m=\frac{3.76\times {10}^5}{3.34\times {10}^5}=1.1\text{kg}\end{array}$

The amount of ice needed is more than tea, which would result in diluted tea. So, it would be better to let cool at room temperature.

Conclusion:

Minimum amount of ice needed is $m=1.1\text{kg}$ .

Yes, it is better to let cool the tea at room temperature before adding ice.